Subgroups — Q₈×C₉×H(3,𝔽₃) #14517833

Order 1944.

606 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×H(3,𝔽₃)	C₂	1	3	yes	maximal
2	C₃×Q₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²×Q₈×C₉	C₃	1	4	yes	maximal, complemented
4	Q₈×C₃²⋊C₉	C₃	1	8	yes	maximal, complemented
5	C₂×C₉×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	3	yes	—
7	C₃²×C₄×C₉	C₂×C₃	1	12	yes	—
8	C₄×C₃²⋊C₉	C₂×C₃	1	24	yes	—
9	C₉×H(3,𝔽₃)	Q₈	1	1	yes	characteristic, direct product factor, Sylow
10	C₃³×Q₈	C₃²	1	4	yes	—
11	Q₈×H(3,𝔽₃)	C₉	1	9	yes	direct product factor
12	C₃×Q₈×C₉	—	3	12	no	—
13	C₃×Q₈×C₉	C₃²	1	8	yes	—
14	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
15	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
16	C₂×C₃²×C₉	C₂²×C₃	1	4	yes	—
17	C₂×C₃²⋊C₉	C₂²×C₃	1	8	yes	—
18	C₃³×C₄	C₂×C₃²	1	12	yes	—
19	C₄×H(3,𝔽₃)	C₂×C₉	1	27	yes	—
20	C₃×C₄×C₉	—	3	36	no	—
21	C₃×C₄×C₉	C₂×C₃²	1	24	yes	—
22	C₃×C₄×C₉	C₂×C₃²	1	3	yes	—
23	C₃×H(3,𝔽₃)	C₃×Q₈	1	1	yes	characteristic
24	C₃²×C₉	C₃×Q₈	1	4	yes	complemented
25	C₃²⋊C₉	C₃×Q₈	1	8	yes	complemented
26	C₃²×Q₈	—	3	4	no	—
27	C₃²×Q₈	—	3	8	no	—
28	C₃²×Q₈	C₃×C₉	1	12	yes	complemented
29	C₃²×Q₈	C₃³	1	1	yes	characteristic
30	Q₈×C₉	—	3	8	no	—
31	Q₈×C₉	H(3,𝔽₃)	1	3	yes	direct product factor
32	C₂×C₃³	C₂²×C₃²	1	4	yes	—
33	C₂×H(3,𝔽₃)	C₂²×C₉	1	9	yes	—
34	C₂×C₃×C₉	—	3	12	no	—
35	C₂×C₃×C₉	C₂²×C₃²	1	8	yes	—
36	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
37	C₃²×C₄	—	3	12	no	—
38	C₃²×C₄	—	3	24	no	—
39	C₃²×C₄	C₂×C₃×C₉	1	36	yes	—
40	C₃²×C₄	C₂×C₃³	1	3	yes	—
41	C₄×C₉	—	3	24	no	—
42	C₄×C₉	C₂×H(3,𝔽₃)	1	9	yes	—
43	C₃³	C₃²×Q₈	1	4	yes	—
44	H(3,𝔽₃)	Q₈×C₉	1	9	yes	direct product factor
45	C₃×C₉	—	3	12	no	—
46	C₃×C₉	C₃²×Q₈	1	8	yes	—
47	C₃×C₉	C₃²×Q₈	1	1	yes	characteristic
48	C₃×Q₈	—	3	12	no	—
49	C₃×Q₈	C₃²×C₉	1	1	yes	characteristic
50	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×Q₈	#737	1	2	yes	—
52	C₂×C₃²	—	3	4	no	—
53	C₂×C₃²	—	3	8	no	—
54	C₂×C₃²	C₂²×C₃×C₉	1	12	yes	—
55	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
56	C₂×C₉	—	3	8	no	—
57	C₂×C₉	C₂²×H(3,𝔽₃)	1	3	yes	—
58	C₃×C₄	—	3	36	no	—
59	C₃×C₄	C₂×C₃²×C₉	1	3	yes	—
60	C₃×C₄	C₂×C₃×H(3,𝔽₃)	1	3	yes	—
61	C₃×C₄	C₂×#737	1	6	yes	—
62	C₃²	—	3	4	no	—
63	C₃²	—	3	8	no	—
64	C₃²	C₃×Q₈×C₉	1	12	yes	complemented
65	C₃²	C₃³×Q₈	1	1	yes	characteristic
66	C₉	—	3	8	no	—
67	C₉	Q₈×H(3,𝔽₃)	1	3	yes	direct product factor
68	Q₈	C₉×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃	—	3	12	no	—
70	C₂×C₃	C₂²×C₃²×C₉	1	1	yes	characteristic
71	C₂×C₃	C₂²×C₃×H(3,𝔽₃)	1	1	yes	characteristic
72	C₂×C₃	C₂²×#737	1	2	yes	—
73	C₄	C₂×C₉×H(3,𝔽₃)	1	3	yes	—
74	C₃	—	3	12	no	—
75	C₃	C₃²×Q₈×C₉	1	1	yes	characteristic
76	C₃	C₃×Q₈×H(3,𝔽₃)	1	1	yes	characteristic
77	C₃	Q₈×#737	1	2	yes	—
78	C₂	C₂²×C₉×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₉	C₂²×C₃²
commutator	C₂×C₃	C₂²×C₃²×C₉
Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×C₃×H(3,𝔽₃)	C₂²×C₃
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃²×Q₈×C₉	C₃	semidirect
4. Q₈×C₃²⋊C₉	C₃	semidirect
9. C₉×H(3,𝔽₃)	Q₈	direct (N × H)
11. Q₈×H(3,𝔽₃)	C₉	direct (N × H)
24. C₃²×C₉	C₃×Q₈	semidirect
25. C₃²⋊C₉	C₃×Q₈	semidirect
28. C₃²×Q₈	C₃×C₉	semidirect
31. Q₈×C₉	H(3,𝔽₃)	direct (N × H)
44. H(3,𝔽₃)	Q₈×C₉	direct (N × H)
64. C₃²	C₃×Q₈×C₉	semidirect
67. C₉	Q₈×H(3,𝔽₃)	direct (N × H)
68. Q₈	C₉×H(3,𝔽₃)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₉×H(3,𝔽₃)	2
C₃×Q₈×H(3,𝔽₃)	3
C₃²×Q₈×C₉	3
Q₈×C₃²⋊C₉	3

Subgroups — Q8×C9×H(3,𝔽3) #14517833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₉×H(3,𝔽₃) #14517833